How many nth-order partial derivatives does a function of two variables have?

Question

Quizplus · Accepted Answer

For a function of two variables, the nth order partial derivative can be found by taking the partial derivative of the (n-1)th order partial derivative with respect to one of the variables. Therefore, the total number of nth order partial derivatives is the number of ways to distribute n derivatives between the two variables without over-counting. This is given by the binomial coefficient (n+2 choose 2), which simplifies to (n+2)(n+1)/2. For n = 0, 1, 2, ..., the values of this expression are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ... The 10th term in this sequence is 66, so the correct answer is D.

How Many Nth-Order Partial Derivatives Does a Function of Two